Minimizing interference for the highway model in Wireless Ad-hoc and Sensor Networks

1. Minimizing interference for the highway model in Wireless Ad-hoc and Sensor Networks Haisheng Tan, Tiancheng, Lou, Francis C.M. Lau, YuexuanWang, Shiteng Chen CS, The University of Hong Kong, Hong Kong, China ITCS, Tsinghua University, Beijing, China Jan. 25th, SOFSEM, 2011

2. Outline • Introduction • Problem Definitions • Minimizing the Average Interference • Minimizing the Maximum Interference • Discussions and Future work • Q & A

3. Introduction • Wireless Ad hoc and Sensor Networks

4. Introduction • Wireless Ad hoc and Sensor Networks • Environmental monitoring, intrusion detection, health care, etc. • Smart Earth (IBM), Sense China …

5. Introduction • Energy !

6. Introduction • Energy ! • Interference

7. Introduction • Energy ! • Interference • Receiver-centric interference transmission radius of u

8. Problem Definitions • the average interference of a graph G • the maximum interference of a graph G

9. Problem Definitions • the average interference of a graph G • the maximum interference of a graph G • Problems: • Given nodes arbitrarily deployed along a 1D line (the highway model) • Connected • Min-Avg or Min-max interference • The optimal solution is actually a spanning tree.

10. Observations

11. Observations • small node degrees

12. Observations • small node degrees • sparse topology

13. Observations • small node degrees • sparse topology • Nearest Neighbor Forest (each node is connected to its nearest neighbor)

14. Observations • small node degrees  • sparse topology  • Nearest Neighbor Forest (each node is connected to its nearest neighbor)  a) b) c)

15. Minimizing the Average Interference • In 2D networks: an asymptotically optimal algorithm with an approximation ratio of O(log n) (Moscibroda et al. 2005)

16. Minimizing the Average Interference • In 2D networks: an asymptotically optimal algorithm with an approximation ratio of O(log n) (Moscibroda et al. 2005) • In the highway model (Our work): a polynomial-time exact algorithm

17. Minimizing the Average Interference • In 2D networks: an asymptotically optimal algorithm with an approximation ratio of O(log n) (Moscibroda et al. 2005) • In the highway model (Our work): 1. No-cross property

18. Minimizing the Average Interference • In 2D networks: an asymptotically optimal algorithm with an approximation ratio of O(log n) (Moscibroda et al. 2005) • In the highway model (Our work): 1. No-cross property when |ac| <=|bc|+|cd| 

19. Minimizing the Average Interference • In the highway model: 2. Calculate the total interference via the interference created by each node

20. Minimizing the Average Interference • In the highway model: 2. Calculate the total interference via the interference created by each node

21. Minimizing the Average Interference • In the highway model: 2. Calculate the total interference via the interference created by each node Independent sub-problems

22. Minimizing the Average Interference • Two questions: • How to divide the whole line into sub-segments • How to connect the nodes inside each segment

23. Minimizing the Average Interference • Two questions: • How to divide the whole line into sub-segments • How to connect the nodes inside each segment • Functions for DP • F(s,t), s<t, which is short for Compute the minimum total interference created by the nodes from s+1 to t-1 , such that

24. Minimizing the Average Interference • Two questions: • How to divide the whole line into sub-segments • How to connect the nodes inside each segment • Functions for DP • F(s,t), s<t, which is short for OR

25. Minimizing the Average Interference • Two questions: • How to divide the whole line into sub-segments • How to connect the nodes inside each segment • Functions for DP • F(s,t), s<t, which is short for OR

26. Minimizing the Average Interference • Functions for DP • G(s,t), s<t Compute the minimum total interference created by nodes from s +1 to t-1, such that

27. Minimizing the Average Interference • Functions for DP • G(s,t), s<t

28. Minimizing the Average Interference • Functions for DP • G(s,t), s<t

29. Minimizing the Average Interference • Functions for DP • G(s,t), s<t • The minimum average interference

30. Minimizing the Average Interference the maximum node degree • Correctness Verified by the brute-force search running in time

31. Minimizing the Average Interference the maximum node degree • Correctness Verified by the brute-force search running in time • Time complexity:

32. Minimizing the Average Interference the maximum node degree • Correctness Verified by the brute-force search running in time • Time complexity: (the numbers are the interference created by the nodes)

33. Minimizing the Average Interference the maximum node degree • Correctness Verified by the brute-force search running in time • Time complexity: (the numbers are the interference created by the nodes) • Can we do better ?? Y! 

34. Minimizing the Maximum Interference • Harder!! • No-cross property: still holds 

35. Minimizing the Maximum Interference • Harder!! • No-cross property: still holds  • Independent sub-segments: not found 

36. Minimizing the Maximum Interference • Harder!! • No-cross property: still holds  • Independent sub-segments: not found  • In 2D networks: NP-hard (Buchin 2008) Bounded in

37. Minimizing the Maximum Interference • Harder!! • No-cross property: still holds  • Independent sub-segments: not found  • In 2D networks: NP-hard (Buchin 2008) Bounded in • In 1D networks: • An appr. with ratio (von Richenbach, et al. 2005) • A sub-exponential-time exact algorithm (Our work)

38. Minimizing the Maximum Interference • Check whether the min-max can be k, where 1<k<n

39. Minimizing the Maximum Interference • Check whether the min-max can be k, where 1<k<n • A skeleton : Record the nodes from s to t that can interfere with nodes outside [s,t] with their transmission radii

40. Minimizing the Maximum Interference • Check whether the min-max can be k, where 1<k<n • A skeleton : Record the nodes from s to t that can interfere with nodes outside [s,t] with their transmission radii

41. Minimizing the Maximum Interference • Check whether the min-max can be k, where 1<k<n • A skeleton : Record the nodes from s to t that can interfere with nodes outside [s,t] with their transmission radii

42. Minimizing the Maximum Interference • Functions: • boolean F*(s,t), which is short for

43. Minimizing the Maximum Interference • Functions: • boolean F*(s,t), which is short for OR

44. Minimizing the Maximum Interference • Functions: • boolean F*(s,t), which is short for OR

45. Minimizing the Maximum Interference • Functions: • boolean G*(s,t)

46. Minimizing the Maximum Interference • Functions: • boolean G*(s,t)

47. Minimizing the Maximum Interference • Functions: • boolean G*(s,t)

48. Minimizing the Maximum Interference • Functions: • boolean G*(s,t) • Check the whole line

49. Minimizing the Maximum Interference • Time complexity • # of the different valid skeletons for a segment from s to t, where s>0 and t<n-1:

50. Minimizing the Maximum Interference • Time complexity • # of the different valid skeletons for a segment from s to t, where s>0 and t<n-1: • Time complexity: